The semiclassical limit of the time dependent Hartree–Fock equation: The Weyl symbol of the solution

Abstract

For a family of solutions to the time dependent Hartree–Fock equation, depending on the semiclassical parameter $h$, we prove that if at the initial time the Weyl symbol of the solution is in $L^1\left({ℝ}^{2n}\right)$ as well as all its derivatives, then this property is true for all time, and we give an asymptotic expansion in powers of $h$ of this Weyl symbol. The main term of the asymptotic expansion is a solution to the Vlasov equation, and the error term is estimated in the norm of $L^1\left({ℝ}^{2n}\right)$.